Edit: also, those infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
https://www.alamy.com/portrait-of-cardinal-flavio-chigii-163... https://www.pinterest.com/pin/462252349233672807/
tiny = infinitesimal
huge = infinity
N = number to be bucketed
tiny <= N < 10 --> bucket 1
10 <= N < 20 --> bucket 2
...
X <= N < huge --> last bucket
As for how common they are, we learn about them in any introductory calculus course when defining derivatives. You come across the idea whenever discussing limits, if somewhat obliquely.
If I learned about it in high school math, and again in "real" math courses at my university, I'd say it's pretty standard.
Also, you seem to be conflating "common" with "standard". "standard" is a mathematical term. Infinitesimal are handwavy in standard analysis (epsilon-delta are the rigorous alternative), but exist rigorously in nonstandard analysis.
There are other instances where I've had a need for such a smallest positive number, where logic is simplified as opposed to checking for 0 in a special way. Whether there's an agreed upon term for that, I know where I've found value in programming tasks.
When I need such a thing, it is almost invariably in comparisons, so I am not doing arithmetic with multiple instances of that smallest representable positive number.
Philosophical problems surrounding the perplexing concept of infinity were already hotly debated by the ancient Greeks. Aristotle made an ontological distinction between actual and potential infinities, and argued that actual infinity cannot exist but potential infinity can. This was also the consensus position of later scholars, and became a sticking point in the acceptance of calculus because infinitesimals (and infinite sums of them) were an example of the ontologically questionable actual infinities.
As I mentioned before, standard modern analysis is based on limits, not infinitesimals, and requires no extension of real numbers. Indeed the limit definition of calculus only requires the concept of potential infinities, so philosophers should be able to rest easy! But infinitesimals still occur in our notation which is largely inherited from Leibniz, however. We say that the derivative of y(x) is dy/dx, or the antiderivative of y(x) is ∫ y(x) dx, and while acknowledging that dy and dx are not actual mathematical objects, just syntax, we still do arithmetic on them whenever it's convenient to do so! For example, when we make a change of variables in an integral, we can substitute x = f(t) for some f, and then say dx/dt = f'(t) and "multiply by dt" to get dx = f'(t) dt to figure out what we should put in the place of the "dx" in the integral.
Actual infinitesimal numbers are not dead, either, they're used in a branch of analysis called nonstandard analysis which formalizes them in the logically rigorous manner that is now expected from mathematics.
________
¹ Not that they had a rigorous theory of real numbers, either, that came in the 19th and early 20th century. In fact what we now understand as formal, axiomatized math didn't really exist before the 19th century at all!